Linear Growth Versus Exponential Growth
Logistic Growth Versus Exponential Growth
Compound Growth Versus Exponential Growth
Exponential Brownian Motion
The Mechanism Of Population Doubling
Generations and Population Doublings
What Is Exponential?

External Links
Malthusian Growth Model
- by Steve McKelvey.
Mathematical Modelling in a Real and Complex World - by the Connected Curriculum Project
Exponential Growth and The Rule Of 70
- by EcoFuture 

Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau

Malthus, Thomas Robert, An Essay on the Principle of Population. J. Johnson. 1798. (1st edition) Library of Economics and Liberty.  

Malthus, Thomas Robert, An Essay on the Principle of Population. John Murray. 1826. (6th edition) Library of Economics and Liberty.

The Mechanism Of Population Doubling

Malthus - The Principle Of Population

In England in 1798, as the world population approached 1 billion, Malthus wrote "An Essay On The Principle Of Population". The principle was mathematical in nature, and stated that populations always tend towards exponential growth (and outstrip the available food supply). He used the sound concept of population doubling to demonstrate what he meant. The fact is, Malthus made his case plain enough for variable rates back in 1830 (A Summary View):

"It may be safely asserted, therefore, that population, when unchecked, increases in a geometrical progression of such nature as to double itself every twenty-five years. This statement, of course, refers to the general result, and not to each intermediate step of the progress. Practically, it would sometimes be slower, and sometimes faster."

Malthus typically wrote of human population doubling times averaging 25 years.  Although a widely quoted figure (Darwin himself used it), this was misleading. In fact, later editions of his essay show that Malthus was definitely aware of the variability of population doubling times (see link to 6th edition above, and the quote below).

Fixed Rate Compound Interest = Malthusian Growth Model

Compound interest is a concept most people have heard of, though usually in relation to financial matters and not to populations. Here is an example of compound interest, where a 1% growth rate is applied to a value of 1 (try it using a basic calculator):

Multiply 1.01 by 1 (enter 1, press X, enter 1.01, press =) , and then keep pressing "=" until you get 2. It will take you about 70 presses. This is the effect of a constant 1% compound interest rate. Try this again with1.02 and you get about 35 presses for a 2% compound interest rate. 

In fact, it is possible to prove that any positive growth rate will result in the doubling of your original figure. As the external links (above) show, this growth model (and accompanying mathematical formula) is known as the Malthusian Growth Model, and is frequently applied to populations.

Human Populations - Calculating The Growth Rate

Demographers typically measure human population growth rates as annual growth rates (also known in evolutionary theory as the Malthusian Parameter). Demographers calculate the annual birth rate per 1,000 people, and the annual death rate per 1,000 people. For example a birth rate of 18 per 1,000 people, and a death rate of 8 per 1,000 people, gives a net gain of 10 people per 1,000. This is then expressed as a 1% growth rate. 

So, at a constant annual growth rate of 1% a human population will double roughly every 70 years, and at a constant 2% a population will double roughly every 35 years. You could just as easily calculate the population tripling or quadrupling times, but demographers prefer to use doubling times.

To attain the 25 year population doubling time used by Malthus, a population would have to sustain a growth rate of 2.8%.

Constant Growth Rate versus Variable Growth Rate

Note that a variable positive growth rate (known as variable compound interest) will also result in doubling your population, which is why our global population has been growing exponentially (see below). It is a common fallacy to assume that a constant growth rate is required for exponential growth. This is what Malthus had to say in A Summary View published in 1830:

"The immediate cause of the increase of population is the excess of the births above deaths; and the rate of increase, or the period of doubling, depends upon the proportion which the excess of the births above the deaths bears to the population."

Logic alone should be enough to show that, if a constant 1% growth rate doubles a population in 70 years, and a constant 2% growth rate doubles a population in 35 years, then a population which experiences variable growth rates falling from 2% to 1% will double somewhere between 70 and 35 years. For example:

In fact, any variation of rates between 1% and 2% will result in growth comparable to both. 

As you will see, this explains how our global population of 3 billion in 1960 doubled to 6 billion by 1999 (doubling in 39 years).

The Three Possible Population Growth States

For any given year, a human population can be described to be in one of three possible growth states:

It is worth repeating that Positive Population Growth leads to exponential growth, which allows us to calculate population doubling times. Negative Population Growth ultimately leads to the extinction of the population in question, which is hardly a desirable long term option for humanity.

Given the broad range of possible growth rates for human populations, it would be reasonable to suppose that the state of ZPG is the least likely of the three possible states. That is, it is comparatively unlikely for a population to attain precise equilibrium between birth rates and death rates for any given year, and far more likely to attain NPG or PPG for a given year. 

However, ZPG could be attained through regular oscillation between the NPG and PPG states over the years, such that the net effect is ZPG. This is known as dynamic equilibrium. Whilst this state of affairs is quite common in the natural world, I remain unconvinced that our global human population will attain or maintain such a state of dynamic equilibrium. Take 2 populations of 1 billion each:

Years elapsed  ========>

70 140 210 280
1% NPG  
- 1 billion
495 million 245 million 121 million 60 million
1 % PPG 
- 1 billion
2 billion 4 billion 8 billion 16 billion
Total 2.495 billion 4.245 billion 8.121 billion 16.060 billion

Therefore, if a population of only 1 billion people continues to grow at a modest 1%, ZPG will not be achieved. Assuming a total population of 6 billion, if 5 billion of them experienced rapid NPG (perhaps reaching 50 million within 100 years), the 1 billion growing at 1 % will quickly replace these losses. Don't put your faith in NPG.

See The Cassandra Prediction - Exploding the ZPG Myth for more.

So, that leaves Positive Population Growth, which leads to population doubling.

Exponential Series

 The use of population doubling gives us the standard exponential series for population doubling:

Non- scientific notation: 1 2 4 8 16 32 64 128 256 512 1024
Scientific notation: 20 21    22    23    24    25    26    27    28  29 210

The missing information in the above table is a timeline. The fact is that variable population doubling times allow us to be flexible in projecting future population growth.

A Well Known Rule Of Thumb - The Rule Of 70

As a rule of thumb, if you divide the annual percentage increase of human population growth into 70 then you get the projected doubling rate. 

Note that the projected population doubling rate should not be confused with the actual population doubling rate. The projected, or future, doubling rate is based upon a sustained annual growth rate as just described, and is easily calculated. The actual, or historical, doubling rate, is measured as how long (in years) it actually took the population to double to its current number (or, how long ago was the population exactly half what it is today?).

Because the current trend is currently toward a decreasing population growth rate, the actual doubling rates these days are typically slower than the projected doubling rates given in the past. If the trend were towards an increasing population growth rate then the actual doubling rates would be typically faster than the projected doubling rates given in the past. None of this invalidates the usefulness or validity of using projected doubling rates to discuss our demographic future.

A Rough Measurement Scale

Using the table below, imagine the first one thousand Homo sapiens sapiens 100,000 years ago in Africa. In 10 doubles they reached over one million (1,024,000) so, rounding down, now imagine them at the 1 space for millions. It's estimated that it took them until 10,000 BC (12,000 years ago) to reach a global population of around 10 million, thus requiring between 3 and 4 more doubles. As 8 million is closer to the estimate of 10 million than 16 million, let's call it 13 doubles in 88,000 years, or an average doubling rate for our hunter gatherer ancestors of approximately 6,769 years.

Thousands 1 2 4 8 16 32 64 128 256 512 1024
Millions 1 2 4 8 16 32 64 128 256 512 1024
Billions 1 2 4 8 16 32 64 128 256 512 1024

(Note: Round 1024 thousand down to 1 million, and 1024 million down to 1 billion). 
See my article A New Malthusian Scale for a more precise measurement scale for measuring populations, similar to the measurement scale used for computers.

Then the first civilizations appeared and we began to use agriculture...

By 1600AD our population was around 500 million, close to the figure of 512 million. That's about 6 doubles in 11,600 years, making a doubling rate of approximately 1,933 years. These are just rough figures, but good enough to prove that things were definitely speeding up! From 500 million in 1600AD, we doubled to 1 billion in 1800AD in just 200 years!

People optimistically point out that Earth's population is growing at just under 2%, but it is slowing down ...

However, as Paul Ehrlich states "The Population Explosion", the projected population doubling rate has slowed from a peak of 33 years in the 1960's (at 2.1 %) to 39 years in 1990 (at 1.8%) - Erhlich (1990, p.16).

If you don't like what Paul Ehrlich has to say, consider these actual doubling rates based on milestones in our human population published by the United Nations Population Fund (1998):-

From 1 billion in 1800 to 2 billion in 1930 - doubled in 130 years (average growth rate = 0.54%)
From 2 billion in 1930 to 4 billion in 1974 - doubled in 44 years (average growth rate = 1.75%)
From 3 billion in 1960 to 6 billion in 1999 - doubled in 39 years (average growth rate = 1.79%)

To me, that's not much of a slowdown! Still, take a look at the Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau. They show a slow but comfortable decline in the global growth rate from 1.29% in 1999 (which gives a predicted doubling time of just over 54 years) to less than 0.5% by 2050. I hope they're right.

However, even assuming a meagre projected population growth rate of 0.5% per annum, your population will double approximately every 140 years. If this rate is sustained for only 1,400 years (a long time for a person, but not for humanity), you will convert 6 billion people (6,000,000,000) into over 6 trillion in just ten doublings! So, it is very unlikely that Earth can sustain even a low 0.5% growth rate.

Replication Bomb

Population doubling should scare anyone that has understood it correctly! Population doubling is real, and I believe its not going to stop, only slow down. On the other hand, if we did manage to control our population growth on Earth (and forego a life amongst the stars), I think our human spirit would ultimately die inside.

In his book A River Out Of Eden British zoologist Richard Dawkins refers to life a replication bomb (Dawkins, 1995)

"We humans are an extremely important manifestation of the replication bomb, because it is through us - through our brains, our symbolic culture and our technology - that the explosion may proceed to the next stage and reverberate through deep space."

Universal Laws of Couttsian Growth Model

All this leads to the following universal laws defined by the Couttsian Growth Model (for both constant and variable rates of compound interest):

However, in practice, populations only very rarely experience constant growth (so in reality the above universal laws will apply to populations which are all experiencing variable rates of growth). Also:

In summary, all populations of all species experience growth and shrinkage based upon a model of exponential growth founded on positive and negative rates of compound interest which typically vary over time. Given the (unfortunate) universal definition of exponential growth (which requires a constant rate of growth) my term for population growth based on variable rate compound interest is Couttsian Growth. This behaviour is described by the Couttsian Growth Model, which is proposed as a scientific law of population growth.

Imagine a weighing scale. Year by year (assuming human population growth) put negative growth rates on the left, and positive growth rates on the right. As soon as one side "outweighs" the other by 70, you have a population doubling (if the positive rates "win") or a population halving (if the negative rates "win"). To find out how this works in more detail, including what happens when we mix negative and positive rates of growth (where the rate is allowed to vary), see The Scales Of 70. The actual doubling or halving time is simply the count of years is took before the scales tipped one way or the other. This is dynamic equilibrium in action!

For an accurate model of real-world population growth using natural logarithms, see The Scales of e

For a real-world example, take another look at the Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau. Or look at the same information country by country via CIA World Fact Book page for Populations Ranking.



Dawkins, Richard, A River Out Of Eden. Phoenix. 1995

Ehrlich, Paul R.&  Anne H, The Population Explosion. Touchstone, Simon & Schuster. 1990.

United Nations Population Fund. Press Release OBV/53 POP/675. 9 July 1998

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Copyright 2001 David A. Coutts
Last modified: 08 November, 2009